Not totally correct but close.
We can fill into Safari using our Safari extension. Chrome can also support this but they must explicitly build in support for 1Password using our App Extension:
https://github.com/AgileBits/onepassword-app-extension
I don't use Chrome myself, and Safari works extremely well in my use. However, if you're a Chrome user you can send in a suggestion to Google to see if they'll build in support for it. It shouldn't take them more than a few hours, and in fact many developers who have contacted us after adding support said it took them a matter of minutes.
For browsers that do not support 1Password's extension then yes, you would need to copy and paste from the application to do what you ask.